Invert knife tool cut or other tool to use
I would extrude a single vertex around, building the model up instead of cutting it out.

&#9096; Ctrl LMB can be used to extrude and position geometry in one step, or E can be used for a two step extrude and move.





That said you might get a similar result much more easily if you auto-trace the image into a vector SVG with a program such as inkscape, then import the SVG into blender.